Two closed quantum systems are in the same state when they where prepared using the same macroscopic (classical), surrounding. Technically this means that the results of measuring a complete set of communting measurements are the same for both systems.
Postulate 1. To any closed quantum system is associated a ray in a Hilbert space (complex vector space with inner product).
In Dirac's notation vectors in the Hilbert space are denoted as \(|\psi\rangle\).
The state of a closed quantum system changes in time according to a unitary transformation. The state \(|\psi (t')\rangle\) at time \(t'\) is related to the state \(|\psi (t)\rangle\) the system at time \(t\) by a unitary operator \(U\) which depends only on the times \(t'\) and \(t\), \[ |\psi (t')\rangle = U(t', t)|\psi (t)\rangle\,. \]
Equivalently, infinitesimal time evolution of the state of a closed quantum system is described by the Schrödinger equation \[ \mathrm{i}\hbar\frac{\partial}{\partial t}|\psi (t)\rangle = H(t)|\psi (t)\rangle\,, \] where \(H(t)\) is a is a self-adjoint operator, called the Hamiltonian of the system.
Measurement is an irreversible process during which information about the measured system is obtained and is rendered sufficiently long-living and stable, that is "objective".
Postulate 3. Measurements are described by a collection \(\{A_a\}\) of measurement operators acting on the state space of the system being measured. If the normalized state of the quantum system immediately before the measurement is \(|\psi\rangle\) then the probability that result \(a\) occurs is \[ p(a)=\langle\psi|A^{\dagger}_a A_a |\psi\rangle\,. \] Given that outcome \(a\) occurred, the state of the quantum system after the measurement becomes \[ |\psi'\rangle = \frac{A_a |\psi\rangle}{\sqrt{\langle\psi|A^{\dagger}_a A_a |\psi\rangle}}\,. \] The measurement operators satisfy the completeness equation \[ \sum_a A^{\dagger}_a A_a = I\,. \]
As a particular case, a projective measurement is described by an observable, \(A\), a self-adjoint operator on the state space of the measured system. The possible outcomes of the measurement are the eigenvalues \(a\) of the observable. The measurement leaves the system in the corresponding eigenstate of \(A\).
A self-adjoint operator \(A\) can be expressed as \[ A =\sum_a a P_a\,. \] Here \(a\) is an eigenvalue of \(A\), and \(P_a\) is the corresponding orthogonal projector onto the space of eigenvectors with eigenvalue \(a\). Projectors satisfy \[ \begin{aligned} P_a P_b &= \delta_{a,b} P_a\,, \\ P^{\dagger}_a & = P_a\,. \end{aligned} \]
If the quantum state before to the measurement is \(|\psi\rangle\), then the outcome \(a\) is obtained with probability \[ p(a) = \langle\psi|P_a|\psi\rangle \] and the state of the quantum system after the measurement becomes \[ |\psi'\rangle = \frac{P_a|\psi\rangle}{\Vert P_a|\psi\rangle\Vert}\,. \]
The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. If the Hilbert space of system \(A\) is \(\mathcal{H}_A\) and the Hilbert space of system \(B\) is \(\mathcal{H}_B\), then the Hilbert space of the composite systems \(AB\) is the tensor product \(\mathcal{H}_A \otimes \mathcal{H}_B\).